
Re: The ambiguity of 0^0 on N
Posted:
Sep 19, 2013 10:24 PM


On Thursday, September 19, 2013 7:36:57 PM UTC4, fom wrote: > On 9/19/2013 1:41 PM, Dan Christensen wrote: > > > On Thursday, September 19, 2013 1:16:44 PM UTC4, Peter Percival wrote: > > >> Dan Christensen wrote: > > >> > > >> > > >> > > >>>> There is a simple definition of exponentiation on N, based on > > >> > > >>>> > > >> > > >>>> cardinal arithmetic. It doesn't have any special cases, but > > >> > > >>>> > > >> > > >>>> 0^0 = 1 falls out of it quite naturally: > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> A^B = {f  f:B>A} > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> In English, the cardinal number of set A raised to the power > > >> > > >>>> > > >> > > >>>> of the cardinal number of set B is the cardinal number of the > > >> > > >>>> > > >> > > >>>> set of all functions from B to A. No exceptions, no special > > >> > > >>>> > > >> > > >>>> cases. > > >> > > >>>> > > >> > > >>> > > >> > > >>> Thanks, but as with any analogy, it may not be perfect. > > >> > > >> > > >> > > >> It's not an analogy, it's a fact. I've just taken off my bookshelves > > >> > > >> the first set theory book that came to hand. It's Kunen's Set Theory, > > >> > > >> an Introduction to Independence Proof. On page 31 he defines cardinal > > >> > > >> exponentiation just as Michael F. Stemper does! You know, don't you, > > >> > > >> that the finite cardinals are the same as the natural numbers? > > >> > > > > > > '^' is being used in quite a different way here that may not be perfectly analogous to that usually defined on N. > > > > > > > WRONG!
Then it should be easy to prove 0^0=1 using only natural number arithmetic. How about it?
Dan Download my DC Proof 2.0 software at http://www.dcproof.com

